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1: 4.31 Special Values and Limits
§4.31 Special Values and Limits
Table 4.31.1: Hyperbolic functions: values at multiples of 1 2 π i .
z 0 1 2 π i π i 3 2 π i
4.31.1 lim z 0 sinh z z = 1 ,
4.31.2 lim z 0 tanh z z = 1 ,
4.31.3 lim z 0 cosh z - 1 z 2 = 1 2 .
2: 4.4 Special Values and Limits
§4.4(iii) Limits
4.4.13 lim x x - a ln x = 0 , a > 0 ,
4.4.14 lim x 0 x a ln x = 0 , a > 0 ,
4.4.19 lim n ( ( k = 1 n 1 k ) - ln n ) = γ = 0.57721 56649 01532 86060 ,
3: 4.17 Special Values and Limits
Table 4.17.1: Trigonometric functions: values at multiples of 1 12 π .
θ sin θ cos θ tan θ csc θ sec θ cot θ
4.17.1 lim z 0 sin z z = 1 ,
4.17.2 lim z 0 tan z z = 1 .
4.17.3 lim z 0 1 - cos z z 2 = 1 2 .
4: 10.30 Limiting Forms
§10.30(i) z 0
5: 20.13 Physical Applications
In the singular limit τ 0 + , the functions θ j ( z | τ ) , j = 1 , 2 , 3 , 4 , become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …
6: 29.5 Special Cases and Limiting Forms
§29.5 Special Cases and Limiting Forms
29.5.5 lim k 1 - Ec ν m ( z , k 2 ) Ec ν m ( 0 , k 2 ) = lim k 1 - Es ν m + 1 ( z , k 2 ) Es ν m + 1 ( 0 , k 2 ) = 1 ( cosh z ) μ F ( 1 2 μ - 1 2 ν , 1 2 μ + 1 2 ν + 1 2 1 2 ; tanh 2 z ) , m even,
29.5.6 lim k 1 - Ec ν m ( z , k 2 ) d Ec ν m ( z , k 2 ) / d z | z = 0 = lim k 1 - Es ν m + 1 ( z , k 2 ) d Es ν m + 1 ( z , k 2 ) / d z | z = 0 = tanh z ( cosh z ) μ F ( 1 2 μ - 1 2 ν + 1 2 , 1 2 μ + 1 2 ν + 1 3 2 ; tanh 2 z ) , m odd,
7: 10.7 Limiting Forms
§10.7 Limiting Forms
8: 28.12 Definitions and Basic Properties
However, these functions are not the limiting values of me ± ν ( z , q ) as ν n ( 0 ) . … Again, the limiting values of ce ν ( z , q ) and se ν ( z , q ) as ν n ( 0 ) are not the functions ce n ( z , q ) and se n ( z , q ) defined in §28.2(vi). …
9: 22.5 Special Values
§22.5(ii) Limiting Values of k
Table 22.5.3: Limiting forms of Jacobian elliptic functions as k 0 .
sn ( z , k ) sin z cd ( z , k ) cos z dc ( z , k ) sec z ns ( z , k ) csc z
Table 22.5.4: Limiting forms of Jacobian elliptic functions as k 1 .
sn ( z , k ) tanh z cd ( z , k ) 1 dc ( z , k ) 1 ns ( z , k ) coth z
10: 10.52 Limiting Forms
§10.52 Limiting Forms